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We propose a novel, connectivity-oriented loss function for training deep convolutional networks to reconstruct network-like structures, like roads and irrigation canals, from aerial images. The main idea behind our loss is to express the connectivity of roads, or canals, in terms of disconnections that they create between background regions of the image. In simple terms, a gap in the predicted road causes two background regions, that lie on the opposite sides of a ground truth road, to touch in prediction. Our loss function is designed to prevent such unwanted connections between background regions, and therefore close the gaps in predicted roads. It also prevents predicting false positive roads and canals by penalizing unwarranted disconnections of background regions. In order to capture even short, dead-ending road segments, we evaluate the loss in small image crops. We show, in experiments on two standard road benchmarks and a new data set of irrigation canals, that convnets trained with our loss function recover road connectivity so well, that it suffices to skeletonize their output to produce state of the art maps. A distinct advantage of our approach is that the loss can be plugged in to any existing training setup without further modifications.